Fenced enclosures for containing horses or other types of livestock are commonly referred to as corrals or pens (large corrals are referred to as arenas). Such enclosures typically are constructed from multiple, interconnected fence sections (commonly referred to as panels). Each fence section typically is constructed from multiple, horizontally-spaced, upright posts and multiple, horizontally extending, vertically spaced, tubular rails welded to the upright posts. Consequently, large openings or spaces in the fence sections are defined between the upright posts and the horizontal rails. A drawback of this type of enclosure is that an animal and/or a person riding an animal can be seriously injured if the animal accidentally inserts a leg through an opening in a fence section. Another drawback is that when training an animal inside the enclosure, the animal can become distracted by activities outside of the enclosure, which slows or inhibits the training process.